It is so frustrating when your child argues with you and walking away or blowing your top may seem like great two options. But it won't solve anything for either for you. If your child is very young, a listening ear usually goes a long way. Once they are done, you can put forward your perspective. Because arguments are not really time bound. With teenagers, the problem is usually silence or quiet anger. Sometimes it is explosive anger. Again listening usually calms the child. Once they are done, put forth reasons why you are concerned rather than simply saying "You can't do this or that". Children are always more open to suggestions when they feel it is coming from a place of concern rather than a place of authority.
Remember when you argue with your child, you do have a choice to decide how to react and reach a solution, rather than an impasse.